Bounds.cpp

#include <iostream>

#include "Bounds.h"
#include "IRVisitor.h"
#include "IR.h"
#include "IROperator.h"
#include "IREquality.h"
#include "Simplify.h"
#include "IRPrinter.h"
#include "Util.h"
#include "Var.h"
#include "Debug.h"
#include "ExprUsesVar.h"
#include "IRMutator.h"
#include "CSE.h"

namespace Halide {
namespace Internal {

using std::make_pair;
using std::map;
using std::vector;
using std::string;
using std::pair;

namespace {
int static_sign(Expr x) {
    if (is_positive_const(x)) {
        return 1;
    } else if (is_negative_const(x)) {
        return -1;
    } else {
        Expr zero = make_zero(x.type());
        if (equal(const_true(), simplify(x > zero))) {
            return 1;
        } else if (equal(const_true(), simplify(x < zero))) {
            return -1;
        }
    }
    return 0;
}
}

Expr find_constant_bound(Expr e, Direction d) {
    // We look through casts, so we only handle ops that can't
    // overflow. E.g. if A >= a and B >= b, then you can't assume that
    // (A + B) >= (a + b) in a world with overflow.
    if (is_const(e)) {
        return e;
    } else if (const Min *min = e.as<Min>()) {
        Expr a = find_constant_bound(min->a, d);
        Expr b = find_constant_bound(min->b, d);
        if (a.defined() && b.defined()) {
            return simplify(Min::make(a, b));
        } else if (a.defined() && d == Direction::Upper) {
            return a;
        } else if (b.defined() && d == Direction::Upper) {
            return b;
        }
    } else if (const Max *max = e.as<Max>()) {
        Expr a = find_constant_bound(max->a, d);
        Expr b = find_constant_bound(max->b, d);
        if (a.defined() && b.defined()) {
            return simplify(Max::make(a, b));
        } else if (a.defined() && d == Direction::Lower) {
            return a;
        } else if (b.defined() && d == Direction::Lower) {
            return b;
        }
    } else if (const Cast *cast = e.as<Cast>()) {
        Expr a = find_constant_bound(cast->value, d);
        if (a.defined()) {
            return simplify(Cast::make(cast->type, a));
        }
    }
    return Expr();
}


class Bounds : public IRVisitor {
public:
    Interval interval;
    Scope<Interval> scope;
    const FuncValueBounds &func_bounds;

    Bounds(const Scope<Interval> *s, const FuncValueBounds &fb) :
        func_bounds(fb) {
        scope.set_containing_scope(s);
    }
private:

    // Compute the intrinsic bounds of a function.
    void bounds_of_func(string name, int value_index, Type t) {
        // if we can't get a good bound from the function, fall back to the bounds of the type.
        bounds_of_type(t);

        pair<string, int> key = make_pair(name, value_index);

        FuncValueBounds::const_iterator iter = func_bounds.find(key);

        if (iter != func_bounds.end()) {
            if (iter->second.has_lower_bound()) {
                interval.min = iter->second.min;
            }
            if (iter->second.has_upper_bound()) {
                interval.max = iter->second.max;
            }
        }
    }

    void bounds_of_type(Type t) {
        t = t.element_of();
        if ((t.is_uint() || t.is_int()) && t.bits() <= 16) {
            interval = Interval(t.min(), t.max());
        } else {
            interval = Interval::everything();
        }
    }

    using IRVisitor::visit;

    void visit(const IntImm *op) {
        interval = Interval::single_point(op);
    }

    void visit(const UIntImm *op) {
        interval = Interval::single_point(op);
    }

    void visit(const FloatImm *op) {
        interval = Interval::single_point(op);
    }

    void visit(const Cast *op) {

        op->value.accept(this);
        Interval a = interval;

        if (a.is_single_point(op->value)) {
            interval = Interval::single_point(op);
            return;
        }

        Type to = op->type.element_of();
        Type from = op->value.type().element_of();

        if (a.is_single_point()) {
            interval = Interval::single_point(Cast::make(to, a.min));
            return;
        }

        // If overflow is impossible, cast the min and max. If it's
        // possible, use the bounds of the destination type.
        bool could_overflow = true;
        if (to.can_represent(from) || to.is_float()) {
            could_overflow = false;
        } else if (to.is_int() && to.bits() >= 32) {
            // If we cast to an int32 or greater, assume that it won't
            // overflow. Signed 32-bit integer overflow is undefined.
            could_overflow = false;
        } else if (a.is_bounded() && from.can_represent(to)) {
            // The other case to consider is narrowing where the
            // bounds of the original fit into the narrower type. We
            // can only really prove that this is the case if they're
            // constants, so try to make the constants first.

            Expr lower_bound = find_constant_bound(a.min, Direction::Lower);
            Expr upper_bound = find_constant_bound(a.max, Direction::Upper);

            if (lower_bound.defined() && upper_bound.defined()) {
                // Cast them to the narrow type and back and see if
                // they're provably unchanged.
                Expr test =
                    (cast(from, cast(to, lower_bound)) == lower_bound &&
                     cast(from, cast(to, upper_bound)) == upper_bound);
                if (can_prove(test)) {
                    could_overflow = false;
                    // Relax the bounds to the constants we found. Not
                    // strictly necessary, but probably helpful to
                    // keep the expressions small.
                    a = Interval(lower_bound, upper_bound);
                }
            }
        }

        if (!could_overflow) {
            // Start with the bounds of the narrow type.
            bounds_of_type(from);
            // If we have a better min or max for the arg use that.
            if (a.has_lower_bound()) interval.min = a.min;
            if (a.has_upper_bound()) interval.max = a.max;
            // Then cast those bounds to the wider type.
            if (interval.has_lower_bound()) interval.min = Cast::make(to, interval.min);
            if (interval.has_upper_bound()) interval.max = Cast::make(to, interval.max);
        } else {
            // This might overflow, so use the bounds of the destination type.
            bounds_of_type(to);
        }
    }

    void visit(const Variable *op) {
        if (scope.contains(op->name)) {
            interval = scope.get(op->name);
        } else if (op->type.is_vector()) {
            // Uh oh, we need to take the min/max lane of some unknown vector. Treat as unbounded.
            bounds_of_type(op->type);
        } else {
            interval = Interval::single_point(op);
        }
    }

    void visit(const Add *op) {
        op->a.accept(this);
        Interval a = interval;
        op->b.accept(this);
        Interval b = interval;

        if (a.is_single_point(op->a) && b.is_single_point(op->b)) {
            interval = Interval::single_point(op);
        } else if (a.is_single_point() && b.is_single_point()) {
            interval = Interval::single_point(a.min + b.min);
        } else {
            interval = Interval::everything();
            if (a.has_lower_bound() && b.has_lower_bound()) {
                interval.min = a.min + b.min;
            }
            if (a.has_upper_bound() && b.has_upper_bound()) {
                interval.max = a.max + b.max;
            }

            // Check for overflow for (u)int8 and (u)int16
            if (!op->type.is_float() && op->type.bits() < 32) {
                if (interval.has_upper_bound()) {
                    Expr no_overflow = (cast<int>(a.max) + cast<int>(b.max) == cast<int>(interval.max));
                    if (!can_prove(no_overflow)) {
                        bounds_of_type(op->type);
                        return;
                    }
                }
                if (interval.has_lower_bound()) {
                    Expr no_overflow = (cast<int>(a.min) + cast<int>(b.min) == cast<int>(interval.min));
                    if (!can_prove(no_overflow)) {
                        bounds_of_type(op->type);
                        return;
                    }
                }
            }
        }
    }

    void visit(const Sub *op) {
        op->a.accept(this);
        Interval a = interval;
        op->b.accept(this);
        Interval b = interval;

        if (a.is_single_point(op->a) && b.is_single_point(op->b)) {
            interval = Interval::single_point(op);
        } else if (a.is_single_point() && b.is_single_point()) {
            interval = Interval::single_point(a.min - b.min);
        } else {
            interval = Interval::everything();
            if (a.has_lower_bound() && b.has_upper_bound()) {
                interval.min = a.min - b.max;
            }
            if (a.has_upper_bound() && b.has_lower_bound()) {
                interval.max = a.max - b.min;
            }

            // Check for overflow for (u)int8 and (u)int16
            if (!op->type.is_float() && op->type.bits() < 32) {
                if (interval.has_upper_bound()) {
                    Expr no_overflow = (cast<int>(a.max) - cast<int>(b.min) == cast<int>(interval.max));
                    if (!can_prove(no_overflow)) {
                        bounds_of_type(op->type);
                        return;
                    }
                }
                if (interval.has_lower_bound()) {
                    Expr no_overflow = (cast<int>(a.min) - cast<int>(b.max) == cast<int>(interval.min));
                    if (!can_prove(no_overflow)) {
                        bounds_of_type(op->type);
                        return;
                    }
                }
            }

            // Check underflow for uint
            if (op->type.is_uint() &&
                interval.has_lower_bound() &&
                !can_prove(b.max <= a.min)) {
                bounds_of_type(op->type);
            }
        }
    }

    void visit(const Mul *op) {

        op->a.accept(this);
        Interval a = interval;

        op->b.accept(this);
        Interval b = interval;

        // Move constants to the right
        if (a.is_single_point() && !b.is_single_point()) {
            std::swap(a, b);
        }

        if (a.is_single_point(op->a) && b.is_single_point(op->b)) {
            interval = Interval::single_point(op);
        } else if (a.is_single_point() && b.is_single_point()) {
            interval = Interval::single_point(a.min * b.min);
        } else if (b.is_single_point()) {
            Expr e1 = a.has_lower_bound() ? a.min * b.min : a.min;
            Expr e2 = a.has_upper_bound() ? a.max * b.min : a.max;
            if (is_zero(b.min)) {
                interval = b;
            } else if (is_positive_const(b.min) || op->type.is_uint()) {
                interval = Interval(e1, e2);
            } else if (is_negative_const(b.min)) {
                interval = Interval(e2, e1);
            } else if (a.is_bounded()) {
                // Sign of b is unknown
                Expr cmp = b.min >= make_zero(b.min.type().element_of());
                interval = Interval(select(cmp, e1, e2), select(cmp, e2, e1));
            } else {
                interval = Interval::everything();
            }
        } else if (a.is_bounded() && b.is_bounded()) {
            interval = Interval::nothing();
            interval.include(a.min * b.min);
            interval.include(a.min * b.max);
            interval.include(a.max * b.min);
            interval.include(a.max * b.max);
        } else {
            interval = Interval::everything();
        }

        if (op->type.bits() < 32 && !op->type.is_float()) {
            if (a.is_bounded() && b.is_bounded()) {
                // Try to prove it can't overflow
                Expr test1 = (cast<int>(a.min) * cast<int>(b.min) == cast<int>(a.min * b.min));
                Expr test2 = (cast<int>(a.min) * cast<int>(b.max) == cast<int>(a.min * b.max));
                Expr test3 = (cast<int>(a.max) * cast<int>(b.min) == cast<int>(a.max * b.min));
                Expr test4 = (cast<int>(a.max) * cast<int>(b.max) == cast<int>(a.max * b.max));
                if (!can_prove(test1 && test2 && test3 && test4)) {
                    bounds_of_type(op->type);
                }
            } else {
                bounds_of_type(op->type);
            }
        }

    }

    void visit(const Div *op) {
        op->a.accept(this);
        Interval a = interval;

        op->b.accept(this);
        Interval b = interval;

        if (!b.is_bounded()) {
            interval = Interval::everything();
        } else if (a.is_single_point(op->a) && b.is_single_point(op->b)) {
            interval = Interval::single_point(op);
        } else if (can_prove(b.min == b.max)) {
            Expr e1 = a.has_lower_bound() ? a.min / b.min : a.min;
            Expr e2 = a.has_upper_bound() ? a.max / b.max : a.max;
            if (is_positive_const(b.min) || op->type.is_uint()) {
                interval = Interval(e1, e2);
            } else if (is_negative_const(b.min)) {
                interval = Interval(e2, e1);
            } else if (a.is_bounded()) {
                // Sign of b is unknown. Note that this might divide
                // by zero, but only in cases where the original code
                // divides by zero.
                Expr cmp = b.min > make_zero(b.min.type().element_of());
                interval = Interval(select(cmp, e1, e2), select(cmp, e2, e1));
            } else {
                interval = Interval::everything();
            }
        } else if (a.is_bounded()) {
            // if we can't statically prove that the divisor can't span zero, then we're unbounded
            int min_sign = static_sign(b.min);
            int max_sign = static_sign(b.max);
            if (min_sign != max_sign || min_sign == 0 || max_sign == 0) {
                interval = Interval::everything();
            } else {
                // Divisor is either strictly positive or strictly
                // negative, so we can just take the extrema.
                interval = Interval::nothing();
                interval.include(a.min / b.min);
                interval.include(a.max / b.min);
                interval.include(a.min / b.max);
                interval.include(a.max / b.max);
            }
        } else {
            interval = Interval::everything();
        }
    }

    void visit(const Mod *op) {
        op->a.accept(this);
        Interval a = interval;

        op->b.accept(this);
        if (!interval.is_bounded()) {
            return;
        }
        Interval b = interval;

        if (a.is_single_point(op->a) && b.is_single_point(op->b)) {
            interval = Interval::single_point(op);
            return;
        }

        Type t = op->type.element_of();

        if (a.is_single_point() && b.is_single_point()) {
            interval = Interval::single_point(a.min % b.min);
        } else {
            // Only consider B (so A can be unbounded)
            if (b.max.type().is_uint() || (b.max.type().is_int() && is_positive_const(b.min))) {
                // If the RHS is a positive integer, the result is in [0, max_b-1]
                interval = Interval(make_zero(t), b.max - make_one(t));
            } else if (b.max.type().is_int()) {
                // mod is always positive
                // x % [4,10] -> [0,9]
                // x % [-8,-3] -> [0,7]
                // x % [-8, 10] -> [0,9]
                interval = Interval(make_zero(t), Max::make(abs(b.min), abs(b.max)) - make_one(t));
            } else {
                // The floating point version has the same sign rules,
                // but can reach all the way up to the original value,
                // so there's no -1.
                interval = Interval(make_zero(t), Max::make(abs(b.min), abs(b.max)));
            }
        }
    }

    void visit(const Min *op) {
        op->a.accept(this);
        Interval a = interval;

        op->b.accept(this);
        Interval b = interval;

        if (a.is_single_point(op->a) && b.is_single_point(op->b)) {
            interval = Interval::single_point(op);
        } else {
            interval = Interval(Interval::make_min(a.min, b.min),
                                Interval::make_min(a.max, b.max));
        }
    }


    void visit(const Max *op) {
        op->a.accept(this);
        Interval a = interval;

        op->b.accept(this);
        Interval b = interval;

        if (a.is_single_point(op->a) && b.is_single_point(op->b)) {
            interval = Interval::single_point(op);
        } else {
            interval = Interval(Interval::make_max(a.min, b.min),
                                Interval::make_max(a.max, b.max));
        }
    }

    void visit(const EQ *op) {
        bounds_of_type(op->type);
    }

    void visit(const NE *op) {
        bounds_of_type(op->type);
    }

    void visit(const LT *op) {
        bounds_of_type(op->type);
    }

    void visit(const LE *op) {
        bounds_of_type(op->type);
    }

    void visit(const GT *op) {
        bounds_of_type(op->type);
    }

    void visit(const GE *op) {
        bounds_of_type(op->type);
    }

    void visit(const And *op) {
        bounds_of_type(op->type);
    }

    void visit(const Or *op) {
        bounds_of_type(op->type);
    }

    void visit(const Not *op) {
        bounds_of_type(op->type);
    }

    void visit(const Select *op) {
        op->true_value.accept(this);
        if (!interval.is_bounded()) {
            return;
        }
        Interval a = interval;

        op->false_value.accept(this);
        if (!interval.is_bounded()) {
            return;
        }
        Interval b = interval;

        bool const_scalar_condition =
            (op->condition.type().is_scalar() &&
             !expr_uses_vars(op->condition, scope));

        if (a.min.same_as(b.min)) {
            interval.min = a.min;
        } else if (const_scalar_condition) {
            interval.min = select(op->condition, a.min, b.min);
        } else {
            interval.min = Interval::make_min(a.min, b.min);
        }

        if (a.max.same_as(b.max)) {
            interval.max = a.max;
        } else if (const_scalar_condition) {
            interval.max = select(op->condition, a.max, b.max);
        } else {
            interval.max = Interval::make_max(a.max, b.max);
        }
    }

    void visit(const Load *op) {
        op->index.accept(this);
        if (interval.is_single_point()) {
            // If the index is const we can return the load of that index
            Expr load_min =
                Load::make(op->type.element_of(), op->name, interval.min, op->image, op->param);
            interval = Interval::single_point(load_min);
        } else {
            // Otherwise use the bounds of the type
            bounds_of_type(op->type);
        }
    }

    void visit(const Ramp *op) {
        // Treat the ramp lane as a free variable
        string var_name = unique_name('t');
        Expr var = Variable::make(op->base.type(), var_name);
        Expr lane = op->base + var * op->stride;
        scope.push(var_name, Interval(make_const(var.type(), 0),
                                      make_const(var.type(), op->lanes-1)));
        lane.accept(this);
        scope.pop(var_name);
    }

    void visit(const Broadcast *op) {
        op->value.accept(this);
    }

    void visit(const Call *op) {
        // If the args are const we can return the call of those args
        // for pure functions. For other types of functions, the same
        // call in two different places might produce different
        // results (e.g. during the update step of a reduction), so we
        // can't move around call nodes.
        std::vector<Expr> new_args(op->args.size());
        bool const_args = true;
        for (size_t i = 0; i < op->args.size() && const_args; i++) {
            op->args[i].accept(this);
            if (interval.is_single_point()) {
                new_args[i] = interval.min;
            } else {
                const_args = false;
            }
        }

        Type t = op->type.element_of();

        if (t.is_handle()) {
            interval = Interval::everything();
            return;
        }

        if (const_args &&
            (op->call_type == Call::PureExtern ||
             op->call_type == Call::Image)) {
            Expr call = Call::make(t, op->name, new_args, op->call_type,
                                   op->func, op->value_index, op->image, op->param);
            interval = Interval::single_point(call);
        } else if (op->is_intrinsic(Call::abs)) {
            Interval a = interval;
            interval.min = make_zero(t);
            if (a.is_bounded()) {
                if (equal(a.min, a.max)) {
                    interval = Interval::single_point(Call::make(t, Call::abs, {a.max}, Call::PureIntrinsic));
                } else if (op->args[0].type().is_int() && op->args[0].type().bits() >= 32) {
                    interval.max = Max::make(Cast::make(t, -a.min), Cast::make(t, a.max));
                } else {
                    a.min = Call::make(t, Call::abs, {a.min}, Call::PureIntrinsic);
                    a.max = Call::make(t, Call::abs, {a.max}, Call::PureIntrinsic);
                    interval.max = Max::make(a.min, a.max);
                }
            } else {
                // If the argument is unbounded on one side, then the max is unbounded.
                interval.max = Interval::pos_inf;
            }
        } else if (op->is_intrinsic(Call::likely) ||
                   op->is_intrinsic(Call::likely_if_innermost)) {
            assert(op->args.size() == 1);
            op->args[0].accept(this);
        } else if (op->is_intrinsic(Call::return_second)) {
            assert(op->args.size() == 2);
            op->args[1].accept(this);
        } else if (op->is_intrinsic(Call::if_then_else)) {
            assert(op->args.size() == 3);
            // Probably more conservative than necessary
            Expr equivalent_select = Select::make(op->args[0], op->args[1], op->args[2]);
            equivalent_select.accept(this);
        } else if (op->is_intrinsic(Call::shift_left) ||
                   op->is_intrinsic(Call::shift_right) ||
                   op->is_intrinsic(Call::bitwise_and)) {
            Expr simplified = simplify(op);
            if (!equal(simplified, op)) {
                simplified.accept(this);
            } else {
                // Just use the bounds of the type
                bounds_of_type(t);
            }
        } else if (op->args.size() == 1 && interval.is_bounded() &&
                   (op->name == "ceil_f32" || op->name == "ceil_f64" ||
                    op->name == "floor_f32" || op->name == "floor_f64" ||
                    op->name == "round_f32" || op->name == "round_f64" ||
                    op->name == "exp_f32" || op->name == "exp_f64" ||
                    op->name == "log_f32" || op->name == "log_f64")) {
            // For monotonic, pure, single-argument functions, we can
            // make two calls for the min and the max.
            interval = Interval(
                Call::make(t, op->name, {interval.min}, op->call_type,
                           op->func, op->value_index, op->image, op->param),
                Call::make(t, op->name, {interval.max}, op->call_type,
                           op->func, op->value_index, op->image, op->param));

        } else if (op->is_intrinsic(Call::extract_buffer_min) ||
                   op->is_intrinsic(Call::extract_buffer_max)) {
            // Bounds query results should have perfect nesting. Their
            // max over a loop is just the same bounds query call at
            // an outer loop level. This requires that the query is
            // also done at the outer loop level so that the buffer
            // arg is still valid, which it is, so it is.
            //
            // TODO: There should be an assert injected in the inner
            // loop to check perfect nesting.
            interval = Interval(
                Call::make(Int(32), Call::extract_buffer_min, op->args, Call::PureIntrinsic),
                Call::make(Int(32), Call::extract_buffer_max, op->args, Call::PureIntrinsic));
        } else if (op->is_intrinsic(Call::memoize_expr)) {
            internal_assert(op->args.size() >= 1);
            op->args[0].accept(this);
        } else if (op->is_intrinsic(Call::trace_expr)) {
            // trace_expr returns argument 4
            internal_assert(op->args.size() >= 5);
            op->args[4].accept(this);
        } else if (op->call_type == Call::Halide) {
            bounds_of_func(op->name, op->value_index, op->type);
        } else {
            // Just use the bounds of the type
            bounds_of_type(t);
        }
    }

    void visit(const Let *op) {
        op->value.accept(this);
        Interval val = interval;

        // We'll either substitute the values in directly, or pass
        // them in as variables and add an outer let (to avoid
        // combinatorial explosion).
        Interval var;
        string min_name = op->name + ".min";
        string max_name = op->name + ".max";

        if (val.has_lower_bound()) {
            if (is_const(val.min)) {
                // Substitute it in
                var.min = val.min;
                val.min = Expr();
            } else {
                var.min = Variable::make(op->value.type().element_of(), min_name);
            }
        }

        if (val.has_upper_bound()) {
            if (is_const(val.max)) {
                // Substitute it in
                var.max = val.max;
                val.max = Expr();
            } else {
                var.max = Variable::make(op->value.type().element_of(), max_name);
            }
        }

        scope.push(op->name, var);
        op->body.accept(this);
        scope.pop(op->name);

        if (interval.has_lower_bound()) {
            if (val.min.defined() && expr_uses_var(interval.min, min_name)) {
                interval.min = Let::make(min_name, val.min, interval.min);
            }
            if (val.max.defined() && expr_uses_var(interval.min, max_name)) {
                interval.min = Let::make(max_name, val.max, interval.min);
            }
        }

        if (interval.has_upper_bound()) {
            if (val.min.defined() && expr_uses_var(interval.max, min_name)) {
                interval.max = Let::make(min_name, val.min, interval.max);
            }
            if (val.max.defined() && expr_uses_var(interval.max, max_name)) {
                interval.max = Let::make(max_name, val.max, interval.max);
            }
        }
    }

    void visit(const LetStmt *) {
        internal_error << "Bounds of statement\n";
    }

    void visit(const AssertStmt *) {
        internal_error << "Bounds of statement\n";
    }

    void visit(const ProducerConsumer *) {
        internal_error << "Bounds of statement\n";
    }

    void visit(const For *) {
        internal_error << "Bounds of statement\n";
    }

    void visit(const Store *) {
        internal_error << "Bounds of statement\n";
    }

    void visit(const Provide *) {
        internal_error << "Bounds of statement\n";
    }

    void visit(const Allocate *) {
        internal_error << "Bounds of statement\n";
    }

    void visit(const Realize *) {
        internal_error << "Bounds of statement\n";
    }

    void visit(const Block *) {
        internal_error << "Bounds of statement\n";
    }
};

Interval bounds_of_expr_in_scope(Expr expr, const Scope<Interval> &scope, const FuncValueBounds &fb) {
    //debug(3) << "computing bounds_of_expr_in_scope " << expr << "\n";
    Bounds b(&scope, fb);
    expr.accept(&b);
    //debug(3) << "bounds_of_expr_in_scope " << expr << " = " << simplify(b.min) << ", " << simplify(b.max) << "\n";
    if (b.interval.has_lower_bound()) {
        internal_assert(b.interval.min.type().is_scalar())
            << "Min of " << expr
            << " should have been a scalar: " << b.interval.min << "\n";
    }
    if (b.interval.has_upper_bound()) {
        internal_assert(b.interval.max.type().is_scalar())
            << "Max of " << expr
            << " should have been a scalar: " << b.interval.max << "\n";
    }
    return b.interval;
}

Region region_union(const Region &a, const Region &b) {
    internal_assert(a.size() == b.size()) << "Mismatched dimensionality in region union\n";
    Region result;
    for (size_t i = 0; i < a.size(); i++) {
        Expr min = Min::make(a[i].min, b[i].min);
        Expr max_a = a[i].min + a[i].extent;
        Expr max_b = b[i].min + b[i].extent;
        Expr max_plus_one = Max::make(max_a, max_b);
        Expr extent = max_plus_one - min;
        result.push_back(Range(simplify(min), simplify(extent)));
        //result.push_back(Range(min, extent));
    }
    return result;
}

void merge_boxes(Box &a, const Box &b) {
    if (b.empty()) {
        return;
    }

    if (a.empty()) {
        a = b;
        return;
    }

    internal_assert(a.size() == b.size());

    bool a_maybe_unused = a.maybe_unused();
    bool b_maybe_unused = b.maybe_unused();

    bool complementary = a_maybe_unused && b_maybe_unused &&
        (equal(a.used, !b.used) || equal(!a.used, b.used));

    for (size_t i = 0; i < a.size(); i++) {
        if (!a[i].min.same_as(b[i].min)) {
            if (a[i].has_lower_bound() && b[i].has_lower_bound()) {
                if (a_maybe_unused && b_maybe_unused) {
                    if (complementary) {
                        a[i].min = select(a.used, a[i].min, b[i].min);
                    } else {
                        a[i].min = select(a.used && b.used, Interval::make_min(a[i].min, b[i].min),
                                          a.used, a[i].min,
                                          b[i].min);
                    }
                } else if (a_maybe_unused) {
                    a[i].min = select(a.used, Interval::make_min(a[i].min, b[i].min), b[i].min);
                } else if (b_maybe_unused) {
                    a[i].min = select(b.used, Interval::make_min(a[i].min, b[i].min), a[i].min);
                } else {
                    a[i].min = Interval::make_min(a[i].min, b[i].min);
                }
                a[i].min = simplify(a[i].min);
            } else {
                a[i].min = Interval::neg_inf;
            }
        }
        if (!a[i].max.same_as(b[i].max)) {
            if (a[i].has_upper_bound() && b[i].has_upper_bound()) {
                if (a_maybe_unused && b_maybe_unused) {
                    if (complementary) {
                        a[i].max = select(a.used, a[i].max, b[i].max);
                    } else {
                        a[i].max = select(a.used && b.used, Interval::make_max(a[i].max, b[i].max),
                                          a.used, a[i].max,
                                          b[i].max);
                    }
                } else if (a_maybe_unused) {
                    a[i].max = select(a.used, Interval::make_max(a[i].max, b[i].max), b[i].max);
                } else if (b_maybe_unused) {
                    a[i].max = select(b.used, Interval::make_max(a[i].max, b[i].max), a[i].max);
                } else {
                    a[i].max = Interval::make_max(a[i].max, b[i].max);
                }
                a[i].max = simplify(a[i].max);
            } else {
                a[i].max = Interval::pos_inf;
            }
        }
    }

    if (a_maybe_unused && b_maybe_unused) {
        if (!equal(a.used, b.used)) {
            a.used = simplify(a.used || b.used);
            if (is_one(a.used)) {
                a.used = Expr();
            }
        }
    } else {
        a.used = Expr();
    }
}

Box box_union(const Box &a, const Box &b) {
    Box result = a;
    merge_boxes(result, b);
    return result;
}

bool boxes_overlap(const Box &a, const Box &b) {
    // If one box is scalar and the other is not, the boxes cannot
    // overlap.
    if (a.size() != b.size() && (a.size() == 0 || b.size() == 0)) {
        return false;
    }

    internal_assert(a.size() == b.size());

    bool a_maybe_unused = a.maybe_unused();
    bool b_maybe_unused = b.maybe_unused();

    // Overlapping requires both boxes to be used.
    Expr overlap = ((a_maybe_unused ? a.used : const_true()) &&
                    (b_maybe_unused ? b.used : const_true()));

    for (size_t i = 0; i < a.size(); i++) {
        if (a[i].has_upper_bound() && b[i].has_lower_bound()) {
            overlap = overlap && b[i].max >= a[i].min;
        }
        if (a[i].has_lower_bound() && b[i].has_upper_bound()) {
            overlap = overlap && a[i].max >= b[i].min;
        }
    }

    // Conservatively, assume they overlap if we can't prove there's no overlap
    return !can_prove(simplify(!overlap));
}

bool box_contains(const Box &outer, const Box &inner) {
    // If the inner box has more dimensions than the outer box, the
    // inner box cannot fit in the outer box.
    if (inner.size() > outer.size()) {
        return false;
    }
    Expr condition = const_true();
    for (size_t i = 0; i < inner.size(); i++) {
        condition = (condition &&
                     (outer[i].min <= inner[i].min) &&
                     (outer[i].max >= inner[i].max));
    }
    if (outer.maybe_unused()) {
        if (inner.maybe_unused()) {
            // inner condition must imply outer one
            condition = condition && ((outer.used && inner.used) == inner.used);
        } else {
            // outer box is conditional, but inner is not
            return false;
        }
    }
    return is_one(simplify(condition));
}

// Compute the box produced by a statement
class BoxesTouched : public IRGraphVisitor {

public:
    BoxesTouched(bool calls, bool provides, string fn, const Scope<Interval> *s, const FuncValueBounds &fb) :
        func(fn), consider_calls(calls), consider_provides(provides), func_bounds(fb) {
        scope.set_containing_scope(s);
    }

    map<string, Box> boxes;

private:

    string func;
    bool consider_calls, consider_provides;
    Scope<Interval> scope;
    const FuncValueBounds &func_bounds;

    using IRGraphVisitor::visit;

    void visit(const Call *op) {
        if (!consider_calls) return;

        // Calls inside of an address_of aren't touched, because no
        // actual memory access takes place.
        if (op->is_intrinsic(Call::address_of)) {
            // Visit the args of the inner call
            internal_assert(op->args.size() == 1);
            const Call *c = op->args[0].as<Call>();

            if (c) {
                for (size_t i = 0; i < c->args.size(); i++) {
                    c->args[i].accept(this);
                }
            } else {
                const Load *l = op->args[0].as<Load>();

                internal_assert(l);
                l->index.accept(this);
            }

            return;
        }

        if (op->is_intrinsic(Call::if_then_else)) {
            assert(op->args.size() == 3);
            // We wrap 'then_case' and 'else_case' inside 'dummy' call since IfThenElse
            // only takes Stmts as arguments.
            Stmt then_case = Evaluate::make(op->args[1]);
            Stmt else_case = Evaluate::make(op->args[2]);
            Stmt equivalent_if = IfThenElse::make(op->args[0], then_case, else_case);
            equivalent_if.accept(this);
            return;
        }

        IRVisitor::visit(op);

        if (op->call_type == Call::Halide ||
            op->call_type == Call::Image) {
            for (Expr e : op->args) {
                e.accept(this);
            }
            if (op->name == func || func.empty()) {
                Box b(op->args.size());
                b.used = const_true();
                for (size_t i = 0; i < op->args.size(); i++) {
                    b[i] = bounds_of_expr_in_scope(op->args[i], scope, func_bounds);
                }
                merge_boxes(boxes[op->name], b);
            }
        }
    }

    class CountVars : public IRVisitor {
        using IRVisitor::visit;

        void visit(const Variable *var) {
            count++;
        }
    public:
        int count;
        CountVars() : count(0) {}
    };

    // We get better simplification if we directly substitute mins
    // and maxes in, but this can also cause combinatorial code
    // explosion. Here we manage this by only substituting in
    // reasonably-sized expressions. We determine the size by
    // counting the number of var nodes.
    bool is_small_enough_to_substitute(Expr e) {
        CountVars c;
        e.accept(&c);
        return c.count < 10;
    }

    template<typename LetOrLetStmt>
    void visit_let(const LetOrLetStmt *op) {
        if (consider_calls) {
            op->value.accept(this);
        }
        Interval value_bounds = bounds_of_expr_in_scope(op->value, scope, func_bounds);

        bool fixed = value_bounds.min.same_as(value_bounds.max);
        value_bounds.min = simplify(value_bounds.min);
        value_bounds.max = fixed ? value_bounds.min : simplify(value_bounds.max);

        if (is_small_enough_to_substitute(value_bounds.min) &&
            (fixed || is_small_enough_to_substitute(value_bounds.max))) {
            scope.push(op->name, value_bounds);
            op->body.accept(this);
            scope.pop(op->name);
        } else {
            string max_name = unique_name('t');
            string min_name = unique_name('t');

            scope.push(op->name, Interval(Variable::make(op->value.type(), min_name),
                                          Variable::make(op->value.type(), max_name)));
            op->body.accept(this);
            scope.pop(op->name);

            for (pair<const string, Box> &i : boxes) {
                Box &box = i.second;
                for (size_t i = 0; i < box.size(); i++) {
                    if (box[i].has_lower_bound()) {
                        if (expr_uses_var(box[i].min, max_name)) {
                            box[i].min = Let::make(max_name, value_bounds.max, box[i].min);
                        }
                        if (expr_uses_var(box[i].min, min_name)) {
                            box[i].min = Let::make(min_name, value_bounds.min, box[i].min);
                        }
                    }
                    if (box[i].has_upper_bound()) {
                        if (expr_uses_var(box[i].max, max_name)) {
                            box[i].max = Let::make(max_name, value_bounds.max, box[i].max);
                        }
                        if (expr_uses_var(box[i].max, min_name)) {
                            box[i].max = Let::make(min_name, value_bounds.min, box[i].max);
                        }
                    }
                }
            }
        }
    }

    void visit(const Let *op) {
        visit_let(op);
    }

    void visit(const LetStmt *op) {
        visit_let(op);
    }

    void visit(const IfThenElse *op) {
        op->condition.accept(this);
        if (expr_uses_vars(op->condition, scope)) {
            if (!op->else_case.defined() || is_no_op(op->else_case)) {
                // Trim the scope down to represent the fact that the
                // condition is true. We only understand certain types
                // of conditions for now.
                Expr c = op->condition;
                const Call *call = c.as<Call>();
                if (call && (call->is_intrinsic(Call::likely) ||
                             call->is_intrinsic(Call::likely_if_innermost))) {
                    c = call->args[0];
                }
                const LT *lt = c.as<LT>();
                const LE *le = c.as<LE>();
                const GT *gt = c.as<GT>();
                const GE *ge = c.as<GE>();
                const EQ *eq = c.as<EQ>();
                Expr a, b;
                if (lt) {a = lt->a; b = lt->b;}
                if (le) {a = le->a; b = le->b;}
                if (gt) {a = gt->a; b = gt->b;}
                if (ge) {a = ge->a; b = ge->b;}
                if (eq) {a = eq->a; b = eq->b;}
                const Variable *var_a = a.as<Variable>();
                const Variable *var_b = b.as<Variable>();

                string var_to_pop;
                if (a.defined() && b.defined() && a.type() == Int(32)) {
                    Expr inner_min, inner_max;
                    if (var_a && scope.contains(var_a->name)) {
                        Interval i = scope.get(var_a->name);

                        // If the original condition is likely, then
                        // the additional trimming of the domain due
                        // to the condition is probably unnecessary,
                        // which means the mins/maxes below should
                        // probably just be the LHS.
                        Interval likely_i = i;
                        if (call && call->is_intrinsic(Call::likely)) {
                            likely_i.min = likely(i.min);
                            likely_i.max = likely(i.max);
                        } else if (call && call->is_intrinsic(Call::likely_if_innermost)) {
                            likely_i.min = likely_if_innermost(i.min);
                            likely_i.max = likely_if_innermost(i.max);
                        }

                        Interval bi = bounds_of_expr_in_scope(b, scope, func_bounds);
                        if (bi.has_upper_bound()) {
                            if (lt) {
                                i.max = min(likely_i.max, bi.max - 1);
                            }
                            if (le || eq) {
                                i.max = min(likely_i.max, bi.max);
                            }
                        }
                        if (bi.has_lower_bound()) {
                            if (gt) {
                                i.min = max(likely_i.min, bi.min + 1);
                            }
                            if (ge || eq) {
                                i.min = max(likely_i.min, bi.min);
                            }
                        }
                        scope.push(var_a->name, i);
                        var_to_pop = var_a->name;
                    } else if (var_b && scope.contains(var_b->name)) {
                        Interval i = scope.get(var_b->name);

                        Interval likely_i = i;
                        if (call && call->is_intrinsic(Call::likely)) {
                            likely_i.min = likely(i.min);
                            likely_i.max = likely(i.max);
                        } else if (call && call->is_intrinsic(Call::likely_if_innermost)) {
                            likely_i.min = likely_if_innermost(i.min);
                            likely_i.max = likely_if_innermost(i.max);
                        }

                        Interval ai = bounds_of_expr_in_scope(a, scope, func_bounds);
                        if (ai.has_upper_bound()) {
                            if (gt) {
                                i.max = min(likely_i.max, ai.max - 1);
                            }
                            if (ge || eq) {
                                i.max = min(likely_i.max, ai.max);
                            }
                        }
                        if (ai.has_lower_bound()) {
                            if (lt) {
                                i.min = max(likely_i.min, ai.min + 1);
                            }
                            if (le || eq) {
                                i.min = max(likely_i.min, ai.min);
                            }
                        }
                        scope.push(var_b->name, i);
                        var_to_pop = var_b->name;
                    }
                }
                op->then_case.accept(this);
                if (!var_to_pop.empty()) {
                    scope.pop(var_to_pop);
                }
            } else {
                // Just take the union over the branches
                op->then_case.accept(this);
                op->else_case.accept(this);
            }
        } else {
            // If the condition is based purely on params, then we'll only
            // ever go one way in a given run, so we should conditionalize
            // the boxes touched on the condition.

            // Fork the boxes touched and go down each path
            map<string, Box> then_boxes, else_boxes;
            then_boxes.swap(boxes);
            op->then_case.accept(this);
            then_boxes.swap(boxes);

            if (op->else_case.defined()) {
                else_boxes.swap(boxes);
                op->else_case.accept(this);
                else_boxes.swap(boxes);
            }

            // Make sure all the then boxes have an entry on the else
            // side so that the merge doesn't skip them.
            for (pair<const string, Box> &i : then_boxes) {
                else_boxes[i.first];
            }

            // Merge
            for (pair<const string, Box> &i : else_boxes) {
                Box &else_box = i.second;
                Box &then_box = then_boxes[i.first];
                Box &orig_box = boxes[i.first];

                if (then_box.maybe_unused()) {
                    then_box.used = then_box.used && op->condition;
                } else {
                    then_box.used = op->condition;
                }

                if (else_box.maybe_unused()) {
                    else_box.used = else_box.used && !op->condition;
                } else {
                    else_box.used = !op->condition;
                }

                merge_boxes(then_box, else_box);
                merge_boxes(orig_box, then_box);
            }
        }
    }

    void visit(const For *op) {
        if (consider_calls) {
            op->min.accept(this);
            op->extent.accept(this);
        }

        Expr min_val, max_val;
        if (scope.contains(op->name + ".loop_min")) {
            min_val = scope.get(op->name + ".loop_min").min;
        } else {
            min_val = bounds_of_expr_in_scope(op->min, scope, func_bounds).min;
        }

        if (scope.contains(op->name + ".loop_max")) {
            max_val = scope.get(op->name + ".loop_max").max;
        } else {
            max_val = bounds_of_expr_in_scope(op->extent, scope, func_bounds).max;
            max_val += bounds_of_expr_in_scope(op->min, scope, func_bounds).max;
            max_val -= 1;
        }

        scope.push(op->name, Interval(min_val, max_val));
        op->body.accept(this);
        scope.pop(op->name);
    }

    void visit(const Provide *op) {
        if (consider_provides) {
            if (op->name == func || func.empty()) {
                Box b(op->args.size());
                for (size_t i = 0; i < op->args.size(); i++) {
                    b[i] = bounds_of_expr_in_scope(op->args[i], scope, func_bounds);
                }
                merge_boxes(boxes[op->name], b);
            }
        }

        if (consider_calls) {
            for (size_t i = 0; i < op->args.size(); i++) {
                op->args[i].accept(this);
            }
            for (size_t i = 0; i < op->values.size(); i++) {
                op->values[i].accept(this);
            }
        }
    }
};

map<string, Box> boxes_touched(Expr e, Stmt s, bool consider_calls, bool consider_provides,
                               string fn, const Scope<Interval> &scope, const FuncValueBounds &fb) {
    // Do calls and provides separately, for better simplification.
    BoxesTouched calls(consider_calls, false, fn, &scope, fb);
    BoxesTouched provides(false, consider_provides, fn, &scope, fb);

    if (consider_calls) {
        if (e.defined()) {
            e.accept(&calls);
        }
        if (s.defined()) {
            s.accept(&calls);
        }
    }
    if (consider_provides) {
        if (e.defined()) {
            e.accept(&provides);
        }
        if (s.defined()) {
            s.accept(&provides);
        }
    }
    if (!consider_calls) {
        return provides.boxes;
    }
    if (!consider_provides) {
        return calls.boxes;
    }

    // Combine the two maps.
    for (pair<const string, Box> &i : provides.boxes) {
        merge_boxes(calls.boxes[i.first], i.second);
    }
    return calls.boxes;
}

Box box_touched(Expr e, Stmt s, bool consider_calls, bool consider_provides,
                string fn, const Scope<Interval> &scope, const FuncValueBounds &fb) {
    map<string, Box> boxes = boxes_touched(e, s, consider_calls, consider_provides, fn, scope, fb);
    internal_assert(boxes.size() <= 1);
    return boxes[fn];
}

map<string, Box> boxes_required(Expr e, const Scope<Interval> &scope, const FuncValueBounds &fb) {
    return boxes_touched(e, Stmt(), true, false, "", scope, fb);
}

Box box_required(Expr e, string fn, const Scope<Interval> &scope, const FuncValueBounds &fb) {
    return box_touched(e, Stmt(), true, false, fn, scope, fb);
}

map<string, Box> boxes_required(Stmt s, const Scope<Interval> &scope, const FuncValueBounds &fb) {
    return boxes_touched(Expr(), s, true, false, "", scope, fb);
}

Box box_required(Stmt s, string fn, const Scope<Interval> &scope, const FuncValueBounds &fb) {
    return box_touched(Expr(), s, true, false, fn, scope, fb);
}

map<string, Box> boxes_provided(Expr e, const Scope<Interval> &scope, const FuncValueBounds &fb) {
    return boxes_touched(e, Stmt(), false, true, "", scope, fb);
}

Box box_provided(Expr e, string fn, const Scope<Interval> &scope, const FuncValueBounds &fb) {
    return box_touched(e, Stmt(), false, true, fn, scope, fb);
}

map<string, Box> boxes_provided(Stmt s, const Scope<Interval> &scope, const FuncValueBounds &fb) {
    return boxes_touched(Expr(), s, false, true, "", scope, fb);
}

Box box_provided(Stmt s, string fn, const Scope<Interval> &scope, const FuncValueBounds &fb) {
    return box_touched(Expr(), s, false, true, fn, scope, fb);
}

map<string, Box> boxes_touched(Expr e, const Scope<Interval> &scope, const FuncValueBounds &fb) {
    return boxes_touched(e, Stmt(), true, true, "", scope, fb);
}

Box box_touched(Expr e, string fn, const Scope<Interval> &scope, const FuncValueBounds &fb) {
    return box_touched(e, Stmt(), true, true, fn, scope, fb);
}

map<string, Box> boxes_touched(Stmt s, const Scope<Interval> &scope, const FuncValueBounds &fb) {
    return boxes_touched(Expr(), s, true, true, "", scope, fb);
}

Box box_touched(Stmt s, string fn, const Scope<Interval> &scope, const FuncValueBounds &fb) {
    return box_touched(Expr(), s, true, true, fn, scope, fb);
}

// Compute interval of all possible function's values (default + specialized values)
Interval compute_pure_function_definition_value_bounds(
        const Definition &def, const Scope<Interval>& scope, const FuncValueBounds &fb, int dim) {

    Interval result = bounds_of_expr_in_scope(def.values()[dim], scope, fb);

    // Pure function might have different values due to specialization.
    // We need to take the union of min and max bounds of all those possible values.
    for (const Specialization &s : def.specializations()) {
        Interval s_interval = compute_pure_function_definition_value_bounds(s.definition, scope, fb, dim);
        result.include(s_interval);
    }
    return result;
}

FuncValueBounds compute_function_value_bounds(const vector<string> &order,
                                              const map<string, Function> &env) {
    FuncValueBounds fb;

    for (size_t i = 0; i < order.size(); i++) {
        Function f = env.find(order[i])->second;
        const vector<string> f_args = f.args();
        for (int j = 0; j < f.outputs(); j++) {
            pair<string, int> key = make_pair(f.name(), j);

            Interval result;

            if (f.is_pure()) {

                // Make a scope that says the args could be anything.
                Scope<Interval> arg_scope;
                for (size_t k = 0; k < f.args().size(); k++) {
                    arg_scope.push(f_args[k], Interval::everything());
                }

                result = compute_pure_function_definition_value_bounds(f.definition(), arg_scope, fb, j);
                // These can expand combinatorially as we go down the
                // pipeline if we don't run CSE on them.
                if (result.has_lower_bound()) {
                    result.min = simplify(common_subexpression_elimination(result.min));
                }

                if (result.has_upper_bound()) {
                    result.max = simplify(common_subexpression_elimination(result.max));
                }

                fb[key] = result;
            }

            debug(2) << "Bounds on value " << j
                     << " for func " << order[i]
                     << " are: " << result.min << ", " << result.max << "\n";
        }
    }

    return fb;
}

void check(const Scope<Interval> &scope, Expr e, Expr correct_min, Expr correct_max) {
    FuncValueBounds fb;
    Interval result = bounds_of_expr_in_scope(e, scope, fb);
    result.min = simplify(result.min);
    result.max = simplify(result.max);
    if (!equal(result.min, correct_min)) {
        internal_error << "In bounds of " << e << ":\n"
                       << "Incorrect min: " << result.min << '\n'
                       << "Should have been: " << correct_min << '\n';
    }
    if (!equal(result.max, correct_max)) {
        internal_error << "In bounds of " << e << ":\n"
                       << "Incorrect max: " << result.max << '\n'
                       << "Should have been: " << correct_max << '\n';
    }
}

void bounds_test() {
    Scope<Interval> scope;
    Var x("x"), y("y");
    scope.push("x", Interval(Expr(0), Expr(10)));

    check(scope, x, 0, 10);
    check(scope, x+1, 1, 11);
    check(scope, (x+1)*2, 2, 22);
    check(scope, x*x, 0, 100);
    check(scope, 5-x, -5, 5);
    check(scope, x*(5-x), -50, 50); // We don't expect bounds analysis to understand correlated terms
    check(scope, Select::make(x < 4, x, x+100), 0, 110);
    check(scope, x+y, y, y+10);
    check(scope, x*y, select(y < 0, y*10, 0), select(y < 0, 0, y*10));
    check(scope, x/(x+y), Interval::neg_inf, Interval::pos_inf);
    check(scope, 11/(x+1), 1, 11);
    check(scope, Load::make(Int(8), "buf", x, Buffer(), Parameter()), make_const(Int(8), -128), make_const(Int(8), 127));
    check(scope, y + (Let::make("y", x+3, y - x + 10)), y + 3, y + 23); // Once again, we don't know that y is correlated with x
    check(scope, clamp(1/(x-2), x-10, x+10), -10, 20);

    check(scope, print(x, y), 0, 10);
    check(scope, print_when(x > y, x, y), 0, 10);

    check(scope, select(y == 5, 0, 3), select(y == 5, 0, 3), select(y == 5, 0, 3));
    check(scope, select(y == 5, x, -3*x + 8), select(y == 5, 0, -22), select(y == 5, 10, 8));
    check(scope, select(y == x, x, -3*x + 8), -22, 10);

    check(scope, cast<int32_t>(abs(cast<int16_t>(x/y))), 0, 32768);
    check(scope, cast<float>(x), 0.0f, 10.0f);

    check(scope, cast<int32_t>(abs(cast<float>(x))), 0, 10);

    // Check some vectors
    check(scope, Ramp::make(x*2, 5, 5), 0, 40);
    check(scope, Broadcast::make(x*2, 5), 0, 20);
    check(scope, Broadcast::make(3, 4), 3, 3);

    // Check some operations that may overflow
    check(scope, (cast<uint8_t>(x)+250), make_const(UInt(8), 0), make_const(UInt(8), 255));
    check(scope, (cast<uint8_t>(x)+10)*20, make_const(UInt(8), 0), make_const(UInt(8), 255));
    check(scope, (cast<uint8_t>(x)+10)*(cast<uint8_t>(x)+5), make_const(UInt(8), 0), make_const(UInt(8), 255));
    check(scope, (cast<uint8_t>(x)+10)-(cast<uint8_t>(x)+5), make_const(UInt(8), 0), make_const(UInt(8), 255));

    // Check some operations that we should be able to prove do not overflow
    check(scope, (cast<uint8_t>(x)+240), make_const(UInt(8), 240), make_const(UInt(8), 250));
    check(scope, (cast<uint8_t>(x)+10)*10, make_const(UInt(8), 100), make_const(UInt(8), 200));
    check(scope, (cast<uint8_t>(x)+10)*(cast<uint8_t>(x)), make_const(UInt(8), 0), make_const(UInt(8), 200));
    check(scope, (cast<uint8_t>(x)+20)-(cast<uint8_t>(x)+5), make_const(UInt(8), 5), make_const(UInt(8), 25));

    check(scope,
          cast<uint16_t>(clamp(cast<float>(x/y), 0.0f, 4095.0f)),
          make_const(UInt(16), 0), make_const(UInt(16), 4095));

    check(scope,
          cast<uint8_t>(clamp(cast<uint16_t>(x/y), cast<uint16_t>(0), cast<uint16_t>(128))),
          make_const(UInt(8), 0), make_const(UInt(8), 128));

    Expr u8_1 = cast<uint8_t>(Load::make(Int(8), "buf", x, Buffer(), Parameter()));
    Expr u8_2 = cast<uint8_t>(Load::make(Int(8), "buf", x + 17, Buffer(), Parameter()));
    check(scope, cast<uint16_t>(u8_1) + cast<uint16_t>(u8_2),
          make_const(UInt(16), 0), make_const(UInt(16), 255*2));

    vector<Expr> input_site_1 = {2*x};
    vector<Expr> input_site_2 = {2*x+1};
    vector<Expr> output_site = {x+1};

    Image<int32_t, 1> in(10);
    Buffer in_buf(in, "input");

    Stmt loop = For::make("x", 3, 10, ForType::Serial, DeviceAPI::Host,
                          Provide::make("output",
                                        {Add::make(Call::make(in_buf, input_site_1),
                                                   Call::make(in_buf, input_site_2))},
                                        output_site));

    map<string, Box> r;
    r = boxes_required(loop);
    internal_assert(r.find("output") == r.end());
    internal_assert(r.find("input") != r.end());
    internal_assert(equal(simplify(r["input"][0].min), 6));
    internal_assert(equal(simplify(r["input"][0].max), 25));
    r = boxes_provided(loop);
    internal_assert(r.find("output") != r.end());
    internal_assert(equal(simplify(r["output"][0].min), 4));
    internal_assert(equal(simplify(r["output"][0].max), 13));

    Box r2({Interval(Expr(5), Expr(19))});
    merge_boxes(r2, r["output"]);
    internal_assert(equal(simplify(r2[0].min), 4));
    internal_assert(equal(simplify(r2[0].max), 19));

    std::cout << "Bounds test passed" << std::endl;
}

}
}